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Abstract

This study aims to derive improved forms of classical integral inequalities for differentiable functions possessing the property of arithmetic–harmonic convexity. To achieve this objective, a generalized version of Hölder’s integral inequality together with the Hölder–İşcan inequality is used to establish new inequalities for such functions. These results play an important role in determining bounds for errors arising in the approximation of integrals. The obtained inequalities not only reduce the error bounds associated with previously known integral inequalities, but also provide a framework for deeper mathematical analysis of certain convex functions. Furthermore, to demonstrate the applicability of the theoretical results, the derived inequalities are applied to several mathematical means, including the arithmetic mean, harmonic mean, and logarithmic mean. These applications show that the presented results have both theoretical significance and practical usefulness, providing a basis for further analytical and applied research in the study of integral inequalities.

Keywords

Arithmetic–Harmonically Convex Function Convexity Harmonically Convex Function Hermite–Hadamard Inequality Hölder Inequality

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Hemat, F. M., Baidar , A. W., & Noori, N. (2026). Refinements of Some New Inequalities for Differentiable Arithmetic – Harmonically Convex Functions. Journal of Natural Sciences – Kabul University, 8(4), 213–236. https://doi.org/10.62810/jns.v8i4.456
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